Place aux nouveaux!
This directly translates to "Make way for the new!" and conveys the same sentiment as the English idiom.
You could also say:
* Fini le vieux, place au nouveau! (Finished with the old, make way for the new!)
* On oublie le passé, on se tourne vers l'avenir! (We forget the past, we turn to the future!)
These options are more literal translations, but they capture the meaning of the idiom.